Final answer:
Both equations 2ln(x)=0 and ln(x²)=0 have the same solution, x = 1, since the natural logarithm of 1 is 0. This demonstrates that the given statement is True.
Step-by-step explanation:
The student asked whether the equations 2ln(x)=0 and ln(x²)=0 have the same solutions. To determine the truth of the statement, let's solve each equation separately.
For the first equation, 2ln(x) = 0, we would divide both sides by 2 to isolate ln(x), resulting in ln(x) = 0. Since ln(x) is the natural logarithm of x, the equation states that x must be the number whose natural logarithm is 0. Recalling that the natural logarithm of 1 is always 0 (ln(1) = 0), the solution to the equation is x = 1.
For the second equation, ln(x²) = 0, using properties of logarithms, specifically that ln(a²) = 2ln(a), we can rewrite the equation as 2ln(x) = 0. This simplifies down to ln(x) = 0, which we already have solved as x = 1.
Both equations, therefore, have exactly the same solution: x = 1. This proves that the statement is True.